\(\int x^4 (a+b x) (a c-b c x)^5 \, dx\) [27]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=\frac {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11} \]

[Out]

1/5*a^6*c^5*x^5-2/3*a^5*b*c^5*x^6+5/7*a^4*b^2*c^5*x^7-5/9*a^2*b^4*c^5*x^9+2/5*a*b^5*c^5*x^10-1/11*b^6*c^5*x^11

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=\frac {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11} \]

[In]

Int[x^4*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

(a^6*c^5*x^5)/5 - (2*a^5*b*c^5*x^6)/3 + (5*a^4*b^2*c^5*x^7)/7 - (5*a^2*b^4*c^5*x^9)/9 + (2*a*b^5*c^5*x^10)/5 -
(b^6*c^5*x^11)/11

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a^6 c^5 x^4-4 a^5 b c^5 x^5+5 a^4 b^2 c^5 x^6-5 a^2 b^4 c^5 x^8+4 a b^5 c^5 x^9-b^6 c^5 x^{10}\right ) \, dx \\ & = \frac {1}{5} a^6 c^5 x^5-\frac {2}{3} a^5 b c^5 x^6+\frac {5}{7} a^4 b^2 c^5 x^7-\frac {5}{9} a^2 b^4 c^5 x^9+\frac {2}{5} a b^5 c^5 x^{10}-\frac {1}{11} b^6 c^5 x^{11} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=c^5 \left (\frac {a^6 x^5}{5}-\frac {2}{3} a^5 b x^6+\frac {5}{7} a^4 b^2 x^7-\frac {5}{9} a^2 b^4 x^9+\frac {2}{5} a b^5 x^{10}-\frac {b^6 x^{11}}{11}\right ) \]

[In]

Integrate[x^4*(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

c^5*((a^6*x^5)/5 - (2*a^5*b*x^6)/3 + (5*a^4*b^2*x^7)/7 - (5*a^2*b^4*x^9)/9 + (2*a*b^5*x^10)/5 - (b^6*x^11)/11)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(\frac {x^{5} \left (-315 b^{6} x^{6}+1386 a \,x^{5} b^{5}-1925 a^{2} x^{4} b^{4}+2475 a^{4} x^{2} b^{2}-2310 a^{5} x b +693 a^{6}\right ) c^{5}}{3465}\)	\(61\)
default	\(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\)	\(76\)
norman	\(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\)	\(76\)
risch	\(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\)	\(76\)
parallelrisch	\(\frac {1}{5} a^{6} c^{5} x^{5}-\frac {2}{3} a^{5} b \,c^{5} x^{6}+\frac {5}{7} a^{4} b^{2} c^{5} x^{7}-\frac {5}{9} a^{2} b^{4} c^{5} x^{9}+\frac {2}{5} a \,b^{5} c^{5} x^{10}-\frac {1}{11} b^{6} c^{5} x^{11}\)	\(76\)

[In]

int(x^4*(b*x+a)*(-b*c*x+a*c)^5,x,method=_RETURNVERBOSE)

[Out]

1/3465*x^5*(-315*b^6*x^6+1386*a*b^5*x^5-1925*a^2*b^4*x^4+2475*a^4*b^2*x^2-2310*a^5*b*x+693*a^6)*c^5

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

-1/11*b^6*c^5*x^11 + 2/5*a*b^5*c^5*x^10 - 5/9*a^2*b^4*c^5*x^9 + 5/7*a^4*b^2*c^5*x^7 - 2/3*a^5*b*c^5*x^6 + 1/5*
a^6*c^5*x^5

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=\frac {a^{6} c^{5} x^{5}}{5} - \frac {2 a^{5} b c^{5} x^{6}}{3} + \frac {5 a^{4} b^{2} c^{5} x^{7}}{7} - \frac {5 a^{2} b^{4} c^{5} x^{9}}{9} + \frac {2 a b^{5} c^{5} x^{10}}{5} - \frac {b^{6} c^{5} x^{11}}{11} \]

[In]

integrate(x**4*(b*x+a)*(-b*c*x+a*c)**5,x)

[Out]

a**6*c**5*x**5/5 - 2*a**5*b*c**5*x**6/3 + 5*a**4*b**2*c**5*x**7/7 - 5*a**2*b**4*c**5*x**9/9 + 2*a*b**5*c**5*x*
*10/5 - b**6*c**5*x**11/11

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

-1/11*b^6*c^5*x^11 + 2/5*a*b^5*c^5*x^10 - 5/9*a^2*b^4*c^5*x^9 + 5/7*a^4*b^2*c^5*x^7 - 2/3*a^5*b*c^5*x^6 + 1/5*
a^6*c^5*x^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{11} \, b^{6} c^{5} x^{11} + \frac {2}{5} \, a b^{5} c^{5} x^{10} - \frac {5}{9} \, a^{2} b^{4} c^{5} x^{9} + \frac {5}{7} \, a^{4} b^{2} c^{5} x^{7} - \frac {2}{3} \, a^{5} b c^{5} x^{6} + \frac {1}{5} \, a^{6} c^{5} x^{5} \]

[In]

integrate(x^4*(b*x+a)*(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

-1/11*b^6*c^5*x^11 + 2/5*a*b^5*c^5*x^10 - 5/9*a^2*b^4*c^5*x^9 + 5/7*a^4*b^2*c^5*x^7 - 2/3*a^5*b*c^5*x^6 + 1/5*
a^6*c^5*x^5

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int x^4 (a+b x) (a c-b c x)^5 \, dx=\frac {a^6\,c^5\,x^5}{5}-\frac {2\,a^5\,b\,c^5\,x^6}{3}+\frac {5\,a^4\,b^2\,c^5\,x^7}{7}-\frac {5\,a^2\,b^4\,c^5\,x^9}{9}+\frac {2\,a\,b^5\,c^5\,x^{10}}{5}-\frac {b^6\,c^5\,x^{11}}{11} \]

[In]

int(x^4*(a*c - b*c*x)^5*(a + b*x),x)

[Out]

(a^6*c^5*x^5)/5 - (b^6*c^5*x^11)/11 - (2*a^5*b*c^5*x^6)/3 + (2*a*b^5*c^5*x^10)/5 + (5*a^4*b^2*c^5*x^7)/7 - (5*
a^2*b^4*c^5*x^9)/9